Mathematical Sciences: Non-Locally Convex Spaces and Their Applications in Analysis

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Professor Kalton's research project continues his investigation of mathematical objects encountered at a fairly high level of abstraction in analysis. At the most concrete level of this subject, one is concerned with summing series, integrating functions, solving differential equations, and the like. Moving up a notch in generality, it has been found to be advantageous to amalgamate the analytic objects one is dealing with into what are called Banach spaces (in the most favorable situations) or other sorts of linear spaces. These spaces in turn can be studied either individually, or along the lines of a general theory. The present project deals with spaces that are related to Banach spaces but do not fit strictly within that category. There is for instance a construction called the twisted sum that is most naturally viewed in a much larger category of functional analytic objects than that of Banach spaces. Two of the latter can have a twisted sum that is not even locally convex. Kalton will study decomposition and extension properties defined with respect to twisted sum, and the connections of this notion with interpolation theory. His methods are expected to have repercussions for problems usually approached just from within the Banach space context.